Question

(a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and portrait for the system. (c) From the plot(s) in part (b) determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. $$ d x / d t=x-x^{2}-x y, \quad d y / d t=\frac{1}{2} y-\frac{1}{4} y^{2}-\frac{3}{4} x y $$

Short answer

Question: List the critical points of the given system of differential equations and classify their stability. Answer: The critical points of the system are (0,0), (0,2), (1,0), and (1/2,1/2). Their stability and types are as follows: (0,0) is an unstable node, (0,2) and (1,0) are asymptotically stable, and (1/2,1/2) is an unstable saddle.

Step-by-step solution

Find the critical points (equilibrium solutions)

To find the critical points, we need to solve the following algebraic system: $$dx / dt = x - x^2 - xy = 0$$ $$dy / dt = \frac{1}{2}y - \frac{1}{4}y^2 - \frac{3}{4}xy = 0$$ Let's first find when dx/dt = 0: $$x - x^2 - xy = 0$$ $$x(1 - x - y) = 0$$ This leads to two possibilities: 1. x = 0 2. 1 - x - y = 0 Now let's find when dy/dt = 0: $$\frac{1}{2}y - \frac{1}{4}y^2 - \frac{3}{4}xy = 0$$ $$y(\frac{1}{2} -\frac{1}{4}y - \frac{3}{4}x) = 0$$ This also leads to two possibilities: 1. y = 0 2. \(\frac{1}{2} -\frac{1}{4}y - \frac{3}{4}x = 0\) Now, let's combine the possibilities to find the critical points: 1. x = 0, y = 0 -> (0,0) 2. x = 0, \(\frac{1}{2} -\frac{1}{4}y - \frac{3}{4}x = 0\) -> (0, 2) 3. y = 0, \(1 - x - y = 0\) -> (1, 0) 4. \(\frac{1}{2} -\frac{1}{4}y - \frac{3}{4}x = 0\), \(1 - x - y = 0\) -> Solve the system of equations to find the critical point. For the 4th set of possibilities, we will solve the equations as follows: $$1 - x - y = 0$$ $$\frac{1}{2} -\frac{1}{4}y - \frac{3}{4}x = 0$$ From the first equation, we get the expression for y: $$y = 1 - x$$ Substitute this expression into the second equation: $$\frac{1}{2} -\frac{1}{4}(1 - x) - \frac{3}{4}x = 0$$ $$\frac{1}{2} -\frac{1}{4} +\frac{1}{4}x - \frac{3}{4}x = 0$$ $$\frac{1}{4} - \frac{2}{4}x = 0$$ $$\frac{1}{2} - x = 0$$ From here, we find x = 1/2 and y = 1/2. Hence the 4th critical point is (1/2, 1/2). So, the critical points are (0,0), (0,2), (1,0), and (1/2,1/2).

Use a computer to draw a direction field and portrait for the system

To draw the direction field and the portrait for the system, one can use a computer program like Matlab, Python or online tools like Desmos or Wolfram Alpha. By plotting the direction field, we will be able to visualize the behavior of the system around the critical points.

Determine the stability and classify the critical points

Based on the direction field from the previous step, we can qualitatively analyze the stability of each critical point. 1. (0,0) - unstable node, as trajectories diverge from the point 2. (0,2) - asymptotically stable, as trajectories converge to the point 3. (1,0) - asymptotically stable, as trajectories converge to the point 4. (1/2,1/2) - unstable saddle, as trajectories move away from the point in some directions but converge to the point in others. In conclusion, the four critical points are (0,0), (0,2), (1,0), and (1/2,1/2) and their stability and types are unstable node, asymptotically stable, asymptotically stable, and unstable saddle, respectively.

Key concepts

Equilibrium Solutions

In mathematics, especially in the study of differential equations, equilibrium solutions or critical points are vital for understanding the behavior of a system. An equilibrium solution is a set of values where the derivatives become zero, meaning there is no change over time for those variables. In our system, we have two equations involving derivatives: \(dx/dt = x - x^2 - xy = 0\) and \(dy/dt = \frac{1}{2}y - \frac{1}{4}y^2 - \frac{3}{4}xy = 0\). The first step to finding these equilibrium solutions is to solve these algebraic equations.When solving, we set both equations to zero, which leads to potential static conditions in the variable space. This means if the system starts at this point, it remains there. For our task, we identify four equilibrium solutions:

• (0,0): Here, both variables are zero, representing a fundamental resting state.
• (0,2): This set indicates the system balances with \(y=2\) and \(x\) at zero.
• (1,0): Demonstrating another balance with x equal to one and y nullified.
• (1/2,1/2): A point where both variables share equal value, balancing influencing factors precisely.

Finding these points is crucial for further analysis, as they serve as anchors for predicting system behavior.

Direction Field

A direction field is a useful graphical tool that helps us visualize the behavior of differential equations. It consists of a grid of arrows, each pointing in the direction of the derivative at that specific point in space. When we create a direction field for a system of differential equations, like our example with \(dx/dt\) and \(dy/dt\), it enables us to see how solutions might flow or evolve over time.To sketch a direction field, you typically use a software tool that will compute and display the field for you, such as Python's Matplotlib or Matlab. The direction field helps one understand where the system is tending towards, especially around the critical points or equilibrium solutions. Arrows show if the solution curves, which are the paths traced by the movement of the system, either spiral towards or away from these points.This visual representation is crucial because it provides a comprehensive understanding of the system dynamics, without having to solve the equations analytically. You can see:

• Whether trajectories converge, indicating stability
• If they diverge, indicating instability

Direction fields thus offer a dynamic way to interpret the mathematical system and anticipate the behavior of solutions around critical points.

Stability Analysis

Stability analysis is about determining whether the system will return to an equilibrium state after a small disturbance. For this, we use the direction field and other methods to analyze the nature of the equilibrium solutions. In our context:

• (0,0) - This is an unstable node. Here, any slight shift in the system leads the solution paths away, showcasing instability.
• (0,2) and (1,0) - These are asymptotically stable. Tiny disturbances in these points result in the system eventually returning back to these points as trajectories converge here.
• (1/2,1/2) - This critical point is an unstable saddle. Its instability means that while some trajectories near this point might lead into it, others diverge, based on initial conditions.

Stability analysis is integral in determining the resilience and predictability of dynamic systems. It helps determine which equilibrium solutions are naturally attracted (stable) or repelled (unstable) by solution curves.

Nonlinear Systems

Nonlinear systems are systems where the change of the system's output is not directly proportional to the change of the input. They involve equations where terms can be products or nonlinear functions of the variables.Our example equations \(dx/dt = x - x^2 - xy\) and \(dy/dt = \frac{1}{2}y - \frac{1}{4}y^2 - \frac{3}{4}xy\) are nonlinear due to the squared terms and products of \(x\) and \(y\). Nonlinear systems are complex and can exhibit rich and unexpected behaviors compared to linear systems.Some challenges with nonlinear systems include:

• Unpredictability: Nonlinear systems can behave erratically with slight changes in initial conditions.
• Multiple equilibria: They often have multiple equilibrium points, as seen in our example.
• Varied stability: The stability around each point can differ greatly.

Analyzing nonlinear systems often requires sophisticated methods, both numerically and analytically, due to these complexities. Despite these challenges, understanding nonlinear systems is key, as they more accurately represent most real-world phenomena.